Question

p698 #27 structure at a pet store, toy tennis balls for pets are sold in 3 different sizes. complete the table by calculating the exact volume for each size ball. record the volumes of each tennis ball in terms of π.

size	small	medium	large

| volume (cm³) |
π |
π |
π |

Explanation:

The volume \( V \) of a sphere is given by the formula \( V=\frac{4}{3}\pi r^{3} \), where \( r \) is the radius of the sphere. The radius \( r \) is half of the diameter \( d \), so \( r = \frac{d}{2} \). We will calculate the volume for each size (Small, Medium, Large) using this formula.

Step 1: Calculate the volume of the Small tennis ball

• Diameter of Small ball, \( d = 3 \) cm. So, radius \( r=\frac{3}{2}=1.5 \) cm.
• Using the volume formula for a sphere: \( V=\frac{4}{3}\pi r^{3} \)
• Substitute \( r = 1.5 \) into the formula:

\( V=\frac{4}{3}\pi(1.5)^{3} \)
First, calculate \( (1.5)^{3}=1.5\times1.5\times1.5 = 3.375 \)
Then, \( \frac{4}{3}\times3.375 = 4.5 \)
So, the volume of the Small ball is \( 4.5\pi \) \( \text{cm}^3 \) or we can write it as \( \frac{9}{2}\pi \) (but \( 4.5\pi \) is also correct).

Step 2: Calculate the volume of the Medium tennis ball

• Diameter of Medium ball, \( d = 4.5 \) cm. So, radius \( r=\frac{4.5}{2}=2.25 \) cm.
• Using the volume formula: \( V=\frac{4}{3}\pi r^{3} \)
• Substitute \( r = 2.25 \) into the formula:

\( V=\frac{4}{3}\pi(2.25)^{3} \)
Calculate \( (2.25)^{3}=2.25\times2.25\times2.25 = 11.390625 \)
Then, \( \frac{4}{3}\times11.390625=\frac{45.5625}{3}=15.1875 \)
So, the volume of the Medium ball is \( 15.1875\pi \) \( \text{cm}^3 \) or we can write it as \( \frac{243}{16}\pi \) (since \( 15.1875=\frac{243}{16} \)) or as a fraction with denominator 8: \( 15.1875=\frac{121.5}{8}=\frac{243}{16} \), but decimal form is also acceptable.

Step 3: Calculate the volume of the Large tennis ball

• Diameter of Large ball, \( d = 6.75 \) cm. So, radius \( r=\frac{6.75}{2}=3.375 \) cm.
• Using the volume formula: \( V=\frac{4}{3}\pi r^{3} \)
• Substitute \( r = 3.375 \) into the formula:

\( V=\frac{4}{3}\pi(3.375)^{3} \)
Calculate \( (3.375)^{3}=3.375\times3.375\times3.375 = 38.443359375 \)
Then, \( \frac{4}{3}\times38.443359375=\frac{153.7734375}{3}=51.2578125 \)
So, the volume of the Large ball is \( 51.2578125\pi \) \( \text{cm}^3 \) or we can write it as a fraction. \( 51.2578125=\frac{820.125}{16}=\frac{1640.25}{32}=\frac{3280.5}{64}=\frac{6561}{128} \) (since \( 3.375 = \frac{27}{8} \), so \( (\frac{27}{8})^{3}=\frac{19683}{512} \), then \( \frac{4}{3}\times\frac{19683}{512}=\frac{78732}{1536}=\frac{6561}{128}=51.2578125 \))

Answer:

• Small: \( 4.5\pi \) (or \( \frac{9}{2}\pi \)) \( \text{cm}^3 \)
• Medium: \( 15.1875\pi \) (or \( \frac{243}{16}\pi \)) \( \text{cm}^3 \)
• Large: \( 51.2578125\pi \) (or \( \frac{6561}{128}\pi \)) \( \text{cm}^3 \)

Filling the table:

Size	Diameter (cm)	Volume (\( \text{cm}^3 \))
Medium	4.5	\( 15.1875\pi \)
Large	6.75	\( 51.2578125\pi \)